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1.
Marilyn Breen 《Journal of Geometry》1986,27(2):175-179
We will establish the following improved Krasnosel'skii theorems for the dimension of the kernel of a starshaped set: For each k and d, 0 k d, define f(d,k) = d+1 if k = 0 and f(d,k) = max{d+1,2d–2k+2} if 1 k d.Theorem 1. Let S be a compact, connected, locally starshaped set in Rd, S not convex. Then for a k with 0 k d, dim ker S k if and only if every f(d, k) lnc points of S are clearly visible from a common k-dimensional subset of S.Theorem 2. Let S be a nonempty compact set in Rd. Then for a k with 0 k d, dim ker S k if and only if every f (d, k) boundary points of S are clearly visible from a common k-dimensional subset of S. In each case, the number f(d, k) is best possible for every d and k. 相似文献
2.
Marilyn Breen 《Journal of Geometry》1983,21(1):42-52
Let S be a compact, connected, locally starshaped set in Rd, S not convex. For every point of local nonconvexity q of S, define Aq to be the subset of S from which q is clearly visible via S. Then ker S = {conv Aq: q lnc S}. Furthermore, if every d+1 points of local nonconvexity of S are clearly visible from a common d-dimensional subset of S, then dim ker S = d. 相似文献
3.
Marilyn Breen 《Geometriae Dedicata》1982,13(2):201-213
Let Sø be a bounded connected set in R
2, and assume that every 3 or fewer lnc points of S are clearly visible from a common point of S. Then for some point p in S, the set A{s : s in S and [p, s] S} is nowhere dense in S. Furthermore, when S is open, then S in starshaped. 相似文献
4.
Marilyn Breen 《Archiv der Mathematik》2005,84(3):282-288
Let k and d be fixed integers, 0kd, and let
be a collection of sets in
If every countable subfamily of
has a starshaped intersection, then
is (nonempty and) starshaped as well. Moreover, if every countable subfamily of
has as its intersection a starshaped set whose kernel is at least k-dimensional, then the kernel of
is at least k-dimensional, too. Finally, dual statements hold for unions of sets.Received: 3 April 2004 相似文献
5.
Ambiguous loci of the metric projection onto compact starshaped sets in a Banach space 总被引:1,自引:0,他引:1
LetE be a real Banach space andL(E) the family of all nonempty compact starshaped subsets ofE. Under the Hausdorff distance,L(E) is a complete metric space. The elements of the complement of a first Baire category subset ofL(E) are called typical elements ofL(E). ForXL(E) we denote by
the metrical projection ontoX, i.e. the mapping which associates to eachaE the set of all points inX closest toa. In this note we prove that, ifE is strictly convex and separable with dimE2, then for a typicalXL(E) the map
is not single valued at a dense set of points. Moreover, we show that a typical element ofL(E) has kernel consisting of one point and set of directions dense in the unit sphere ofE. 相似文献
6.
Marilyn Breen 《Journal of Geometry》1989,35(1-2):19-25
Let
and assume that there is a countable collection of lines {L
i
: 1 i} such that (int cl S)
and ((int cl S) S) L
i has one-dimensional Lebesgue measure zero, 1 i. Then every 4 point subset ofS sees viaS a set of positive two-dimensional Lebesgue measure if and only if every finite subset ofS sees viaS such a set. Furthermore, a parallel result holds with two-dimensional replaced by one-dimensional. Finally, setS is finitely starlike if and only if every 5 points ofS see viaS a common point. In each case, the number 4 or 5 is best possible.Supported in part by NSF grant DMS-8705336. 相似文献
7.
Marilyn Breen 《Journal of Geometry》1988,32(1-2):1-12
Let S be a subset of the plane. In case (int cl S) S = , then S is finitely starlike if and only if every 4 points of S see via S a common point. In case (int cl S) S has at most countably many components, each a singleton set, then S is finitely starlike if and only if every 5 points of S see via S a common point. Each of the numbers 4 and 5 is best possible. Examples show that these results fail without suitable restrictions on (int cl S) S. Moreover, a final example shows that if a general Krasnosel'skii number . exists to characterize finitely starlike sets in the plane, then > 9. 相似文献
8.
Marilyn Breen 《Geometriae Dedicata》1991,37(3):317-326
Let be a family of compact starshaped sets in the plane. If every three and every two members of have a union which is connected and simply connected, then {F:F in } is simply connected and nonempty. Of course, if every three and every two members of have a starshaped union, the same result holds.Supported in part by NSF grants DMS-8705336, DMS-8908717 and by a Senior Faculty Summer Research Fellowship, Research Council, University of Oklahoma. 相似文献
9.
Marilyn Breen 《Aequationes Mathematicae》2004,67(3):263-275
Summary.
We establish the following Helly-type result for infinite families
of starshaped sets in
Define the function f on
{1, 2} by
f(1) = 4,
f(2) = 3.
Let
be a fixed positive number, and let
be a uniformly bounded family of compact sets
in the plane. For k = 1, 2, if every
f(k)
(not necessarily distinct) members of
intersect in a starshaped set whose
kernel contains a k-dimensional
neighborhood of radius
, then
is a starshaped set whose kernel is at least
k-dimensional.
The number f(k) is best in each case.
In addition, we present a few results concerning the dimension of
the kernel in an intersection of starshaped sets in
Some of these involve finite families of sets, while others
involve infinite families and make use of the Hausdorff metric. 相似文献
10.
Marilyn Breen 《Journal of Geometry》1990,37(1-2):48-54
For eachk andd, 1kd, definef(d, d)=d+1 andf(d, k)=2d if 1kd–1. The following results are established:Let
be a uniformly bounded collection of compact, convex sets inR
d
. For a fixedk, 1kd, dim {MM in
}k if and only if for some > 0, everyf(d, k) members of
contain a commonk-dimensional set of measure (volume) at least.LetS be a bounded subset ofR
d
. Assume that for some fixedk, 1kd, there exists a countable family of (k–l)-flats {H
i
:i1} inR
d
such that clS S {Hi i 1 } and for eachi1, (clS S) H
i
has (k–1) dimensional measure zero. Every finite subset ofS sees viaS a set of positivek-dimensional measure if and only if for some>0, everyf(d,k) points ofS see viaS a set ofk-dimensional measure at least .The numbers off(d,d) andf(d, 1) above are best possible.Supported in part by NSF grant DMS-8705336. 相似文献
11.
Marilyn Breen 《Journal of Geometry》1999,65(1-2):50-53
Let
be a finite family of compact sets in the plane, and letk be a fixed natural number. If every three (not necessarily distinct) members of
have a union which is simply connected and starshaped viak-paths, then
and
is starshaped viak-paths. Analogous results hold for paths of length at most , > 0, and for staircase paths, although not for staircasek-paths.Supported in part by NSF grant DMS-9504249 相似文献
12.
Marilyn Breen 《Journal of Geometry》1982,19(1):183-196
Let S be an arbitrary nonempty set in Rd. The following results are true for every k, 0kd: the dimension of ker S is at least k if and only if every countable family of boundary points of S is clearly visible from a common k-dimensional neighborhood in S. Similarly, ker S contains a k-dimensional -neighborhood if and only if every countable family of boundary points of S is clearly visible from a common k-dimensional -neighborhood in S.In the plane, we have the following results concerning finitely starlike sets: for S an arbitrary nonempty set in R2, S is finitely starlike if every three points of cl S are clearly visible from a common point of S. In case S –R2 and int cl SS=, then S is finitely starlike if and only if every three points of S are visible from a common point of S. In each case, the number 3 is best possible. 相似文献
13.
Marilyn Breen 《Geometriae Dedicata》1999,75(2):177-186
Let S be an orthogonal polygon in the plane, S simply connected, and let k=2,3. Set S is a union of k sets starshaped via staircase paths if and only if for every F finite, F
bdry S, there is a set G
bdry S arbitrarily close to F and points si,1 ik, (depending on G) such that each point of G is clearly visible from some si. An analogous result holds for a union of 2 sets starshaped via -paths when S is a closed simply connected set in the plane. Each result is best possible. 相似文献
14.
Let S be a subset of R
d
. The set S is said to be an set if and only if for every two points x and y of S, there exists some z S such that [x, z] [z, y] S. Clearly every starshaped set is an set, yet the converse is false and introduces an interesting question: Under what conditions will an set S be almost starshaped; that is, when will there exist a convex subset C of S such that every point of S sees some point of C via SThis paper provides one answer to the question above, and we have the following result: Let S be a closed planar set, S simply connected, and assume that the set Q of points of local nonconvexity of S is finite. If some point p of S see each member of Q via S, then there is a convex subset C of S such that every point of S sees some point of C via S. 相似文献
15.
Marilyn Breen 《Journal of Geometry》1989,35(1-2):14-18
SetS inR
d has propertyK
2 if and only ifS is a finite union ofd-polytopes and for every finite setF in bdryS there exist points c1,c2 (depending onF) such that each point ofF is clearly visible viaS from at least one ci,i = 1,2. The following characterization theorem is established: Let
, d2. SetS is a compact union of two starshaped sets if and only if there is a sequence {S
j
} converging toS (relative to the Hausdorff metric) such that each setS
j satisfies propertyK
2. For
, the sufficiency of the condition above still holds, although the necessity fails. 相似文献
16.
Let f C[a, b]. LetP be a subset ofC[a, b], L b – a be a given real number. We say thatp P is a best approximation tof fromP, with arc length constraintL, ifA[p]
b
a
[1 + (p(x))
2]dx L andp – f q – f for allq P withA[q] L. represents an arbitrary norm onC[a, b]. The constraintA[p] L might be interpreted physically as a materials constraint.In this paper we consider the questions of existence, uniqueness and characterization of constrained best approximations. In addition a bound, independent of degree, is found for the arc length of a best unconstrained Chebyshev polynomial approximation.The work of L. L. Keener is supported by the National Research Council of Canada Grant A8755. 相似文献
17.
M. Katchalski 《Aequationes Mathematicae》1978,17(1):249-254
Denoting by dimA the dimension of the affine hull of the setA, we prove that if {K
i:i T} and {K
i
j
:i T} are two finite families of convex sets inR
n
and if dim {K
i
:i S} = dim {K
i
j
:i S}for eachS T such that|S| n + 1 then dim {K
i
:i T} = dim {K
i
: {i T}}. 相似文献
18.
Victor Chepoi 《Geometriae Dedicata》1996,63(3):321-329
Let P
n
be a union of a finite number of boxes whose intersection graph is a tree. If every two boundary points of P are visible via staircase paths from a common point of P, then P is starshaped via staircase paths. The same result holds true when P is a cubical polyhedron of
n
, which is the geometric realization of some median graph.This generalizes the recent result of M. Breen, J. Geometry, 51 (1994), established for simple rectilinear polygons.Research for this paper was done while the author was visiting the Mathematisches Seminar der Universität Hamburg, on leave from the Universitatea de Stat din Moldova. The author gratefully acknowledges financial support by the Alexander von Humboldt Stifting. 相似文献
19.
Thek-core of the setS
n
is the intersection of the convex hull of all setsA S with ¦SA¦<-k. The Caratheodory number of thek-core is the smallest integerf (d,k) with the property thatx core
kS, S
n
implies the existence of a subsetT S such thatx corekT and ¦T¦f (d, k). In this paper various properties off(d, k) are established.Research of this author was partially supported by Hungarian National Science Foundation grant no. 1812. 相似文献
20.
Péter Komjáth 《Analysis Mathematica》1984,10(4):283-293
. , A
0,A
1,— - lim supA
j
- H,
. , - - . , , ; , , . - . - . 相似文献